The hydrodynamic picture of
quantum mechanics, first developed fully by Bohm following earlier ideas, has
recaptured attention in recent years as an alternative to the conventional
Schrödinger description. The main appeal of Bohm's approach is its formulation
in terms of “trajectories”, which allows a classical-like visualization of
quantum mechanical events. In fact, the Bohmian wavefunction has a form very
closely related to the time-dependent semiclassical approximation. The main
difference is the presence of a “quantum potential”, which is given by the local
curvature of the instantaneous density. The presence of this nonclassical
potential field leads to an interdependence of the quantum trajectories.

The most serious practical difficulty in the use
of quantum trajectories is the need for concurrent evaluation of the quantum
force needed to update the density. In time-dependent semiclassical theory
quantum interference effects arise from cross terms corresponding to distinct
classical trajectories with fixed boundary conditions. Unlike in classical
dynamics, it can be shown that Bohmian trajectories cannot cross in position
space. As a consequence, Bohm's wavefunction consists of a single term. The
repulsive force necessary to prevent crossing events originates in the quantum
potential. Accurate self-consistent determination of the rugged quantum
potential and density poses a numerical challenge and often renders the solution
unstable. Successful calculation of Bohmian trajectories in bound anharmonic
systems has been possible only by combining the hydrodynamic equations with a
direct solution of the Schrödinger equation.

The
hydrodynamic and semiclassical formulations account for quantum mechanical
effects in strikingly different ways. Bohm's expression for an expectation
value can be written in an initial value representation, where the quantum
mechanical phase is entirely absent. By contrast, in the analogous
semiclassical expression off-diagonal phase differences between trajectories in
the forward and backward propagation steps are entirely responsible for quantum
interference. In fact, elimination of such phase differences through
forward-backward, linearization, or similar stationary phase approximations
produces expressions that are incapable of accounting for quantum interference.
It is thus intriguing that Bohm's fully quantum mechanical method can assume a
similar quasiclassical form.

How does the Bohmian method capture quantum mechanical interference in the
absence of phase difference factors associated with multiple-bounce
trajectories? We have addressed this question by examining the quantum
potential and the evolution of Lagrangian fields in a model bound anharmonic
oscillator. Our analysis showed that quantum interference manifests itself
directly as a spatial variation of the density surrounding kinky
trajectories that result from steep forces operating in regions where the
corresponding classical solution exhibits focal points or caustics. The
deviations of quantum trajectories from the underlying classical solutions are
extremely severe in the vicinity of such points, and this behavior represents
the leading source of instability in the Bohmian methodology. These features of
the hydrodynamic approach, which constitute the hallmark of quantum interference
and are ubiquitous in bound nonlinear systems, represent a major source of
instability, making the integration of the Bohmian equations extremely demanding
in such situations. Classical caustics are less common in barrier problems and
entirely absent from the dynamics of quadratic Hamiltonians (except at times
that are multiples of a half period) and thus the numerical difficulties
encountered in such systems are not severe.

We have shown that the quantum force in Bohm's
formulation of quantum mechanics can be related to the stability properties of
the given quantum trajectory. In turn, the evolution of the stability
properties is governed by higher order derivatives of the quantum potential,
leading to an infinite hierarchy of coupled differential equations whose
solution specifies completely all aspects of the dynamics. Neglecting
derivatives of the quantum potential beyond a certain order allows truncation of
the hierarchy, leading to approximate Bohmian trajectories that provide an
accurate description of tunneling phenomena. Use of the method in conjunction
with the quantum initial value representation discussed above allows the use of
Monte Carlo methods for sampling the trajectory initial conditions.

We have also pointed out that the
hydrodynamic formulation of quantum mechanics lends itself as a powerful tool
for solving the diffusion equation. We introduced a wavefunction repartitioning
methodology that prevents imaginary-time trajectory crossing events and thus
leads to stable evolution, overcoming the numerical obstacles that characterize
Bohm's formulation in real time. Use of our approximate technique that focuses
on stability properties to solve Bohm’s equations in imaginary time allows
determination of the energies of low-lying eigenstates from a single
quantum trajectory.

Finally, we have derived a novel fully quantum mechanical formulation of
time correlation functions based on quantum trajectories integrated along a
forward-backward time contour. Unlike its semiclassical analogue, the new
expression involves a smooth integrand amenable to Monte Carlo integration with
a readily available position space density. Further, the quantum potential
governing the motion of the quantum forward-backward trajectories is weak and
well-behaved, facilitating numerical integration.